Properties

Label 4014.2.a.u
Level 4014
Weight 2
Character orbit 4014.a
Self dual yes
Analytic conductor 32.052
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.103354048.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -\beta_{4} + \beta_{5} ) q^{5} + ( 1 - \beta_{4} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -\beta_{4} + \beta_{5} ) q^{5} + ( 1 - \beta_{4} ) q^{7} + q^{8} + ( -\beta_{4} + \beta_{5} ) q^{10} + ( \beta_{3} - \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{5} ) q^{13} + ( 1 - \beta_{4} ) q^{14} + q^{16} + ( \beta_{2} - \beta_{3} ) q^{17} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( -\beta_{4} + \beta_{5} ) q^{20} + ( \beta_{3} - \beta_{4} ) q^{22} + ( 4 - \beta_{1} ) q^{23} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( -\beta_{1} + \beta_{5} ) q^{26} + ( 1 - \beta_{4} ) q^{28} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -2 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{31} + q^{32} + ( \beta_{2} - \beta_{3} ) q^{34} + ( 3 + \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{35} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{37} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{38} + ( -\beta_{4} + \beta_{5} ) q^{40} + ( 4 + \beta_{1} + \beta_{2} ) q^{41} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{43} + ( \beta_{3} - \beta_{4} ) q^{44} + ( 4 - \beta_{1} ) q^{46} + ( 2 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{47} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{49} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{50} + ( -\beta_{1} + \beta_{5} ) q^{52} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{53} + ( 1 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{55} + ( 1 - \beta_{4} ) q^{56} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{59} + ( -4 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{61} + ( -2 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{62} + q^{64} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{65} + ( 4 - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{67} + ( \beta_{2} - \beta_{3} ) q^{68} + ( 3 + \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{70} + ( 3 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{71} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{73} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{74} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{76} + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{77} + ( 3 - 3 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{79} + ( -\beta_{4} + \beta_{5} ) q^{80} + ( 4 + \beta_{1} + \beta_{2} ) q^{82} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{83} + ( -2 + \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{85} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{86} + ( \beta_{3} - \beta_{4} ) q^{88} + ( 2 - 6 \beta_{1} - \beta_{2} - \beta_{5} ) q^{89} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{91} + ( 4 - \beta_{1} ) q^{92} + ( 2 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{94} + ( 1 + 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{95} + ( -3 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 6q^{4} + 2q^{5} + 8q^{7} + 6q^{8} + O(q^{10}) \) \( 6q + 6q^{2} + 6q^{4} + 2q^{5} + 8q^{7} + 6q^{8} + 2q^{10} + 2q^{11} - 2q^{13} + 8q^{14} + 6q^{16} + 2q^{17} - 2q^{19} + 2q^{20} + 2q^{22} + 22q^{23} + 12q^{25} - 2q^{26} + 8q^{28} + 8q^{29} - 12q^{31} + 6q^{32} + 2q^{34} + 20q^{35} - 2q^{38} + 2q^{40} + 28q^{41} + 14q^{43} + 2q^{44} + 22q^{46} + 2q^{47} + 12q^{50} - 2q^{52} + 26q^{53} + 6q^{55} + 8q^{56} + 8q^{58} + 4q^{59} - 6q^{61} - 12q^{62} + 6q^{64} + 16q^{65} + 18q^{67} + 2q^{68} + 20q^{70} + 12q^{71} - 20q^{73} - 2q^{76} + 20q^{77} + 12q^{79} + 2q^{80} + 28q^{82} + 12q^{83} - 10q^{85} + 14q^{86} + 2q^{88} - 2q^{89} - 22q^{91} + 22q^{92} + 2q^{94} + 6q^{95} - 6q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 8 x^{4} + 14 x^{3} + 13 x^{2} - 16 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} + \nu^{3} - 7 \nu^{2} - 6 \nu + 5 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - \nu^{4} - 8 \nu^{3} + 6 \nu^{2} + 13 \nu - 5 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 8 \nu^{3} + 7 \nu^{2} + 13 \nu - 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(6 \beta_{5} - 6 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(8 \beta_{5} - 7 \beta_{4} + \beta_{3} + 7 \beta_{2} + 28 \beta_{1} + 3\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.249769
0.605879
−1.40037
2.34874
−2.35901
2.55499
1.00000 0 1.00000 −2.93762 0 2.50626 1.00000 0 −2.93762
1.2 1.00000 0 1.00000 −2.63291 0 −2.24656 1.00000 0 −2.63291
1.3 1.00000 0 1.00000 −1.03898 0 −0.299718 1.00000 0 −1.03898
1.4 1.00000 0 1.00000 2.51659 0 4.97725 1.00000 0 2.51659
1.5 1.00000 0 1.00000 2.56494 0 2.27923 1.00000 0 2.56494
1.6 1.00000 0 1.00000 3.52797 0 0.783532 1.00000 0 3.52797
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4014.2.a.u yes 6
3.b odd 2 1 4014.2.a.t 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4014.2.a.t 6 3.b odd 2 1
4014.2.a.u yes 6 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(223\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{6} - 2 T_{5}^{5} - 19 T_{5}^{4} + 30 T_{5}^{3} + 110 T_{5}^{2} - 112 T_{5} - 183 \)
\( T_{7}^{6} - 8 T_{7}^{5} + 11 T_{7}^{4} + 36 T_{7}^{3} - 84 T_{7}^{2} + 22 T_{7} + 15 \)
\( T_{11}^{6} - 2 T_{11}^{5} - 22 T_{11}^{4} + 28 T_{11}^{3} + 75 T_{11}^{2} + 42 T_{11} + 7 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{6} \)
$3$ \( \)
$5$ \( 1 - 2 T + 11 T^{2} - 20 T^{3} + 105 T^{4} - 162 T^{5} + 567 T^{6} - 810 T^{7} + 2625 T^{8} - 2500 T^{9} + 6875 T^{10} - 6250 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 8 T + 53 T^{2} - 244 T^{3} + 959 T^{4} - 3142 T^{5} + 8933 T^{6} - 21994 T^{7} + 46991 T^{8} - 83692 T^{9} + 127253 T^{10} - 134456 T^{11} + 117649 T^{12} \)
$11$ \( 1 - 2 T + 44 T^{2} - 82 T^{3} + 922 T^{4} - 1454 T^{5} + 12305 T^{6} - 15994 T^{7} + 111562 T^{8} - 109142 T^{9} + 644204 T^{10} - 322102 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 2 T + 53 T^{2} + 112 T^{3} + 1395 T^{4} + 2548 T^{5} + 22751 T^{6} + 33124 T^{7} + 235755 T^{8} + 246064 T^{9} + 1513733 T^{10} + 742586 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 2 T + 82 T^{2} - 156 T^{3} + 3084 T^{4} - 5034 T^{5} + 67215 T^{6} - 85578 T^{7} + 891276 T^{8} - 766428 T^{9} + 6848722 T^{10} - 2839714 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 2 T + 79 T^{2} + 82 T^{3} + 2853 T^{4} + 1532 T^{5} + 65377 T^{6} + 29108 T^{7} + 1029933 T^{8} + 562438 T^{9} + 10295359 T^{10} + 4952198 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 22 T + 330 T^{2} - 3376 T^{3} + 27572 T^{4} - 177050 T^{5} + 944549 T^{6} - 4072150 T^{7} + 14585588 T^{8} - 41075792 T^{9} + 92347530 T^{10} - 141599546 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 8 T + 91 T^{2} - 558 T^{3} + 4893 T^{4} - 23934 T^{5} + 161495 T^{6} - 694086 T^{7} + 4115013 T^{8} - 13609062 T^{9} + 64362571 T^{10} - 164089192 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 12 T + 163 T^{2} + 1446 T^{3} + 11543 T^{4} + 77210 T^{5} + 461997 T^{6} + 2393510 T^{7} + 11092823 T^{8} + 43077786 T^{9} + 150533923 T^{10} + 343549812 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 146 T^{2} + 16 T^{3} + 9794 T^{4} + 1096 T^{5} + 426551 T^{6} + 40552 T^{7} + 13407986 T^{8} + 810448 T^{9} + 273627506 T^{10} + 2565726409 T^{12} \)
$41$ \( 1 - 28 T + 540 T^{2} - 7068 T^{3} + 74874 T^{4} - 627126 T^{5} + 4445591 T^{6} - 25712166 T^{7} + 125863194 T^{8} - 487133628 T^{9} + 1525910940 T^{10} - 3243973628 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 - 14 T + 246 T^{2} - 2256 T^{3} + 23498 T^{4} - 162252 T^{5} + 1270461 T^{6} - 6976836 T^{7} + 43447802 T^{8} - 179367792 T^{9} + 841025046 T^{10} - 2058118202 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 2 T + 161 T^{2} - 258 T^{3} + 11803 T^{4} - 18340 T^{5} + 608617 T^{6} - 861980 T^{7} + 26072827 T^{8} - 26786334 T^{9} + 785628641 T^{10} - 458690014 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 26 T + 491 T^{2} - 6142 T^{3} + 65029 T^{4} - 553746 T^{5} + 4352587 T^{6} - 29348538 T^{7} + 182666461 T^{8} - 914402534 T^{9} + 3874226171 T^{10} - 10873082818 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 4 T + 178 T^{2} + 18 T^{3} + 12718 T^{4} + 53854 T^{5} + 669709 T^{6} + 3177386 T^{7} + 44271358 T^{8} + 3696822 T^{9} + 2156890258 T^{10} - 2859697196 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 6 T + 145 T^{2} + 358 T^{3} + 13345 T^{4} + 44368 T^{5} + 1081027 T^{6} + 2706448 T^{7} + 49656745 T^{8} + 81259198 T^{9} + 2007646945 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 18 T + 357 T^{2} - 4422 T^{3} + 52655 T^{4} - 514304 T^{5} + 4532217 T^{6} - 34458368 T^{7} + 236368295 T^{8} - 1329973986 T^{9} + 7193950197 T^{10} - 24302251926 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 12 T + 311 T^{2} - 2688 T^{3} + 43470 T^{4} - 315580 T^{5} + 3864635 T^{6} - 22406180 T^{7} + 219132270 T^{8} - 962064768 T^{9} + 7903032791 T^{10} - 21650752212 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 20 T + 432 T^{2} + 5194 T^{3} + 66924 T^{4} + 598592 T^{5} + 5963879 T^{6} + 43697216 T^{7} + 356637996 T^{8} + 2020554298 T^{9} + 12268040112 T^{10} + 41461431860 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 12 T + 404 T^{2} - 4826 T^{3} + 71522 T^{4} - 769226 T^{5} + 7243841 T^{6} - 60768854 T^{7} + 446368802 T^{8} - 2379406214 T^{9} + 15735832724 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 12 T + 427 T^{2} - 4648 T^{3} + 81391 T^{4} - 743694 T^{5} + 8769517 T^{6} - 61726602 T^{7} + 560702599 T^{8} - 2657665976 T^{9} + 20264703067 T^{10} - 47268487716 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 2 T + 145 T^{2} - 98 T^{3} - 675 T^{4} - 113172 T^{5} - 1106865 T^{6} - 10072308 T^{7} - 5346675 T^{8} - 69086962 T^{9} + 9097624945 T^{10} + 11168118898 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 6 T + 283 T^{2} + 2374 T^{3} + 46809 T^{4} + 416110 T^{5} + 5208643 T^{6} + 40362670 T^{7} + 440425881 T^{8} + 2166685702 T^{9} + 25053786523 T^{10} + 51524041542 T^{11} + 832972004929 T^{12} \)
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